Maximum Satisfiability Algorithms Research Articles

N-level Modulo-Based CNF encodings of Pseudo-Boolean constraints for MaxSAT

Many combinatorial problems in various fields can be translated to Maximum Satisfiability (MaxSAT) problems. Although the general problem is $\mathcal {N}\mathcal {P}$ -hard, more and more practical problems may be solved due to the significant effort which has been devoted to the development of efficient solvers. The art of constraints encoding is as important as the art of devising algorithms for MaxSAT. In this paper, we present several encoding methods of pseudo-Boolean constraints into Boolean satisfiability problems in Conjunctive Normal Form (CNF) formula, which are based on the idea of modular arithmetic and only generate auxiliary variables for each unique combination of weights. These techniques are efficient in encoding and solving MaxSAT problems. In particular, our solvers won the partial MaxSAT industrial category from 2010 through 2012 and ranked second in the 2017 main weighted track of the MaxSAT evaluation. We prove the correctness and the pseudo-polynomial space complexity of our encodings and also give a heuristics of the base selection for modular arithmetic. Our experimental results show that our encoding compactly encodes the constraints, and the obtained clauses are efficiently handled by a state-of-the-art SAT solver.

Empirical investigation of stochastic local search for maximum satisfiability

The maximum satisfiability (MAX-SAT) problem is an important NP-hard problem in theory, and has a broad range of applications in practice. Stochastic local search (SLS) is becoming an increasingly popular method for solving MAX-SAT. Recently, a powerful SLS algorithm called CCLS shows efficiency on solving random and crafted MAX-SAT instances. However, the performance of CCLS on solving industrial MAX-SAT instances lags far behind. In this paper, we focus on experimentally analyzing the performance of SLS algorithms for solving industrial MAX-SAT instances. First, we conduct experiments to analyze why CCLS performs poor on industrial instances. Then we propose a new strategy called additive BMS (Best from Multiple Selections) to ease the serious issue. By integrating CCLS and additive BMS, we develop a new SLS algorithm for MAX-SAT called CCABMS, and related experiments indicate the efficiency of CCABMS. Also, we experimentally analyze the effectiveness of initialization methods on SLS algorithms for MAX-SAT, and combine an effective initialization method with CCABMS, resulting in an enhanced algorithm. Experimental results show that our enhanced algorithm performs better than its state-of-the-art SLS competitors on a large number of industrial MAX-SAT instances.

On Using Incremental Encodings in Unsatisfiability-based MaxSAT Solving

In recent years, unsatisfiability-based algorithms have become prevalent as state of the art for solving industrial instances of Maximum Satisfiability (MaxSAT). These algorithms perform a succession of unsatisfiable SAT solver calls until an optimal solution is found. In several of these algorithms, cardinality or pseudo-Boolean constraints are extended between iterations. However, in most cases, the formula provided to the SAT solver in each iteration must be rebuilt and no knowledge is reused from one iteration to the next. This paper describes how to implement incremental unsatisfiability-based algorithms for MaxSAT. In particular, we detail and analyze our implementation of the MSU3 algorithm in the OPENWBO framework that performed remarkably well in the MaxSAT Evaluation of 2014. Furthermore, we also propose to extend incrementality to weighted MaxSAT through an incremental encoding of pseudo-Boolean constraints. The experimental results show that the performance of MaxSAT algorithms can be greatly improved by exploiting the learned information and maintaining the internal state of the SAT solver between iterations. Finally, the proposed incremental encodings of cardinality and pseudo-Boolean constraints are not exclusive for MaxSAT usage and can be used in other domains.

Exploiting subproblem optimization in SAT-based MaxSAT algorithms

The Maximum Satisfiability (MaxSAT) problem is an optimization variant of the Satisfiability (SAT) problem. Several combinatorial optimization problems can be translated into a MaxSAT formula. Among exact MaxSAT algorithms, SAT-based MaxSAT algorithms are the best performing approaches for real-world problems. We have extended the WPM2 algorithm by adding several improvements. In particular, we show that by solving some subproblems of the original MaxSAT instance we can dramatically increase the efficiency of WPM2. This led WPM2 to achieve the best overall results at the international MaxSAT Evaluation 2013 (MSE13) on industrial instances. Then, we present additional techniques and heuristics to further exploit the information retrieved from the resolution of the subproblems. We exhaustively analyze the impact of each improvement what contributes to our understanding of why they work. This architecture allows to convert exact algorithms into efficient incomplete algorithms. The resulting solver had the best results on industrial instances at the incomplete track of the latest international MSE.

An effective variable selection heuristic in SLS for weighted Max-2-SAT

Stochastic local search (SLS) is an appealing method for solving the maximum satisfiability (Max-SAT) problem. This paper proposes a new variable selection heuristic for Max-SAT local search algorithms, which works particularly well for weighted Max-2-SAT instances. Evolving from the recent configuration checking strategy, this new heuristic works in three levels and is called CCTriplex. According to the CCTriplex heuristic, a variable that is both decreasing and configuration changed has the higher priority to be flipped than a decreasing variable, which in turn has the higher priority than a configuration changed variable. The CCTriplex heuristic is used to develop a new SLS algorithm for weighted Max-2-SAT called CCMaxSAT. We evaluate CCMaxSAT on random benchmarks with different densities, and the hand crafted Frb benchmark, as well as weighted Max-2-SAT instances encoded from MaxCut, MaxClique and sports scheduling problems. Compared with the state-of-the-art SLS solver for weighted Max-2-SAT called ITS and the best SLS solver in Max-SAT Evaluation 2012 namely ubcsat-IRoTS, as well as the famous complete solver wMaxSATz, our algorithm CCMaxSAT shows rather good performance on all the benchmarks.

Improving linear search algorithms with model-based approaches for MaxSAT solving

Linear search algorithms have been shown to be particularly effective for solving partial Maximum Satisfiability (MaxSAT) problem instances. These algorithms start by adding a new relaxation variable to each soft clause and solving the resulting formula with a SAT solver. Whenever a Model is found, a new constraint on the relaxation variables is added such that Models with a greater or equal value are excluded. However, if the problem instance has a large number of relaxation variables, then adding a new constraint over these variables can lead to the exploration of a much larger search space. This article proposes new algorithms to improve the performance of linear search algorithms for MaxSAT by using Models found by the SAT solver to partition the relaxation variables. These algorithms add a new constraint on a subset of relaxation variables, thus intensifying the search on that subspace. In addition, the proposed algorithms are further enhanced by using incremental approaches where learned clauses are kept between calls to the SAT solver. Experimental results show that Model-based algorithms can outperform a classic linear search algorithm in several problem instances. Moreover, incremental approaches further improve the performance of linear search and Model-based algorithms for MaxSAT. Overall, Model-based algorithms are competitive with state-of-the-art MaxSAT solvers, and can improve their performance when solving problem instances with a large number of soft clauses.

Iterative and core-guided MaxSAT solving: A survey and assessment

Maximum Satisfiability (MaxSAT) is an optimization version of SAT, and many real world applications can be naturally encoded as such. Solving MaxSAT is an important problem from both a theoretical and a practical point of view. In recent years, there has been considerable interest in developing efficient algorithms and several families of algorithms have been proposed. This paper overviews recent approaches to handle MaxSAT and presents a survey of MaxSAT algorithms based on iteratively calling a SAT solver which are particularly effective to solve problems arising in industrial settings. First, classic algorithms based on iteratively calling a SAT solver and updating a bound are overviewed. Such algorithms are referred to as iterative MaxSAT algorithms. Then, more sophisticated algorithms that additionally take advantage of unsatisfiable cores are described, which are referred to as core-guided MaxSAT algorithms. Core-guided MaxSAT algorithms use the information provided by unsatisfiable cores to relax clauses on demand and to create simpler constraints. Finally, a comprehensive empirical study on non-random benchmarks is conducted, including not only the surveyed algorithms, but also other state-of-the-art MaxSAT solvers. The results indicate that (i) core-guided MaxSAT algorithms in general abort in less instances than classic solvers based on iteratively calling a SAT solver and that (ii) core-guided MaxSAT algorithms are fairly competitive compared to other approaches.

Boolean lexicographic optimization: algorithms & applications

Multi-Objective Combinatorial Optimization (MOCO) problems find a wide range of practical application problems, some of which involving Boolean variables and constraints. This paper develops and evaluates algorithms for solving MOCO problems, defined on Boolean domains, and where the optimality criterion is lexicographic. The proposed algorithms build on existing algorithms for either Maximum Satisfiability (MaxSAT), Pseudo-Boolean Optimization (PBO), or Integer Linear Programming (ILP). Experimental results, obtained on problem instances from haplotyping with pedigrees and software package dependencies, show that the proposed algorithms can provide significant performance gains over state of the art MaxSAT, PBO and ILP algorithms. Finally, the paper also shows that lexicographic optimization conditions are observed in the majority of the problem instances from the MaxSAT evaluations, motivating the development of dedicated algorithms that can exploit lexicographic optimization conditions in general MaxSAT problem instances.